## Orthogonal Laurent polynomials and quadratures on the unit circle and the real half-line

Ruymán Cruz-Barroso and Pablo González-Vera

### Abstract

The purpose of this paper is the computation of quadrature formulas based on Laurent polynomials in two particular situations: the Real Half-Line and the Unit Circle. Comparative results and a connection with the split Levinson algorithm are established. Illustrative numerical examples are approximate integrals of the form $$\int_{-1}^{1} \frac{f(x)}{(x+\lambda)^r} \omega(x) \,dx \;\;,\;\;r=1,2,3,\ldots$$ with $f(x)$ a continuous function on $[-1,1]$, $\omega(x) \geq 0$ a weight function on this interval and $\lambda \in {\bf R} \;\;$ such that $|\lambda|>1$ is required. Here the classical Gaussian quadrature is an extremely slow procedure.

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### Key words

orthogonal Laurent polynomials, L-Gaussian quadrature, Szegő quadrature, three-term recurrence relations, split Levinson algorithm, numerical quadrature.

### AMS subject classifications

41A55, 33C45, 65D30.

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